Related papers: Uniformization of Deligne-Mumford curves

Let $\sigma$ be an automorphism of a field $K$ with fixed field $F$. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras $K[t;\sigma]/fK[t;\sigma]$ obtained…

Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree $p^n$ and the…

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a…

We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which 6 is invertible.…

We study automorphic categories of nilpotent sheaves under degenerations of smooth curves to nodal Deligne-Mumford curves. Our constructions realize affine Hecke operators as the result of bubbling projective lines from marked points. We…

Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms…

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the…

Much of the analysis of F-theory-based Standard Models boils down to computing cohomologies of line bundles on matter curves. By varying parameters one can degenerate such matter curves to singular ones, typically with many nodes, where the…

We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a presentation for M(F) defined in terms of the mapping class groups of the…

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field…

The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider…

O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion…

In the theory of the moduli-stacks of n-pointed stable curves, there are two fundamental functors, contraction and stabilization. These functors are constructed in [4], where they are used to show that the various \bar{M_{g,n}}'s are…

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of…

We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced…

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an…

On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology…

We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the…

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also…